Phase Transitions in Random Graphs

Yesterday, I attended a great talk on Phase Transitions in Random Graphs, the second lecture by visiting scholar Cris Moore of the Santa Fe Institute.

Now, you may be wondering, “Phase Transitions in Random Graphs”? What does that even mean?

Well, I’m glad you asked.

First, “graph” is the technical math term for a network. So we’re talking about networks here, not about random bar charts or something. The most common random graph is the Erdős–Rényi model developed by Paul Erdős and Alfred Rényi. (Interestingly, a similar model was developed separately and simultaneously by Edgar Gilbert who gets none of the credit, but that is a different post.)

The Erdős–Rényi model is simple: you have a set number of vertices and you connect two vertices with an edge with probability p.

Imagine you are a really terrible executive for a new airline company: there are a set number of airports in the world, and you randomly assign direct flights between cities. If you don’t have much start up capital, you might have a low probability of connecting two cities – resulting in a random smattering of flights. So maybe a customer could fly between Boston and New York or between San Francisco and Chicago, but not between Boston and Chicago. If your airline has plenty of capital, though, you might have a high probability of flying between two cities, resulting a connected route allowing a customer to fly from anywhere to anywhere.

The random network is a helpful baseline for understanding what network characteristics are likely to occur “by chance,” but as you may gather from the example above – real networks aren’t random. A new airline would presumably have a strategy for deciding where to fly – focusing on a region and connecting to at least a few major airports.

A phase transition in a network is similar conceptually to a phase transition in a physical system: ice undergoes a phase transition to become a liquid and can undergo another phase transition to become a gas.

A random network undergoes a phase transition when it goes from having lots of disconnected little bits to having a large component.

But when/why does this happen?

Let’s imagine a random network with nodes connected with probability p. In this network,p = k/n where k is a constant and n is the number of nodes in the network. We would then expect each node to have an average degree of k.

So if I’m a random node in this network, I can calculate the average size of the component I’m in. I am one node, connected to k nodes. Since each of those nodes are also connected to k nodes, that makes k^2 nodes connected back to me. This continues outwards as a geometric series. For small k, the geometric series formula tells us that this function will converge at 1 / (1 – k).

So we would expect something wild and crazy to happen when k = 1.

And it does.

This is called the “critical point” of a random network. It is at this point when a network goes from a random collection of disconnected nodes and small components to having a large component. This is the random network’s phase transition.